Mixture and Alligation

Important Concepts and Formulas - Mixture and Alligation1. AlligationAlligation is the rule which enables us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture of a specified price.
2. Mean PriceMean price is the cost price of a unit quantity of the mixture

3. Suppose a container contains $x$ units of a liquid from which $y$ units are taken out and replaced by water. After $n$ operations, quantity of pure liquid
$=x{(1-\frac{y}{x})}^n$ units.

4. Rule of AlligationIf two ingredients are mixed, then
$\left(\frac{\text{Quantity of cheaper}}{\text{Quantity of dearer}}\right)$ $=\left(\frac{\text{CP of dearer - Mean Price}}{\text{Mean price - CP of cheaper}}\right)$

The above formula can be represented with the help of the following diagram which is easier to understand.


=> (Cheaper quantity) : (Dearer quantity) = (d - m) : (m - c)

Proof of Rule of Alligation formula

Let cost price of unit quantity of cheaper ingredient = c
cost price of unit quantity of dearer ingredient = d

Suppose 'a' units of cheaper ingredient and 'b' units of dearer ingredient are mixed.

Total quantity of the mixture = a + b
Total cost price of the mixture = (ac + bd)

cost price of unit quantity of the mixture (mean price) = $\frac{\text{ac + bd}}{\text{a+b}}$

Suppose mean price = m
=> $\frac{\text{ac + bd}}{\text{a+b}}$ = m
=> am + bm = ac + bd
=> am - ac = bd - bm
=> a(m - c) = b(d -m)
=> a/b = (d -m)/(m - c)
=> a : b = (d -m):(m - c)